Here's how it went in my paper.
Equivalence relation แบ่งสมาชิกออกเป็น distinct classes.Claim:
If there are two sets of equivalent elements, and one element from one set is equal to an element from the other set, then the two sets are considered the same class; that is, no different classes have the same element--the classes are distinct.Proof: Supposed that there are two sets of equivalent elements

and

.
Let

, then

(reflexivity).
But

, so

(transitivity),
and

, so

(transitivity).
Since each element of one set is equivalent to all elements of the other set,
the two sets can be merged into a single class.
Thus, two sets cannot be considered two classes if there exists at least one common element.
Therefore, two classes are distinct.